Parameter Optimization Design of the Commutation Circuit of a Hybrid DC-Current-Limiting Circuit Breaker

Abstract

Aimed at the optimization design of the parameters of the commutation circuit of a hybrid DC-current-limiting circuit breaker (HDCCLCB), a parameter selection model considering the short-time withstand of the thyristor and the volume of the commutation circuit is proposed by simplifying the object, and the commutation circuit parameters were preliminarily obtained. In order to verify the correctness of the method of selecting the commutation circuit parameters, the circuit simulation model of the HDCCLCB was built. The experimental platform was built, and the breaking experiment was completed under the fault condition with the current rising rate of 20 A/μs. The correctness of the simulation model and parameter design method was verified by comparing the circuit model simulation results with the experimental results. In order to further optimize the parameters of the commutation circuit, a mathematical model for optimization design was established. Taking the maximum critical breakdown voltage under unit capacitance energy as the objective function, the arcing time before commutating and the current-limiting inductance and capacitance are enumerated to obtain the optimized commutation circuit parameters. By comparing this result with the result of the preliminary design, the objective function is improved by 20.7%, laying a solid foundation for further research and the development of current-limiting circuit breakers for medium-voltage DC systems.

1. Introduction

With the improvement of ship’s electrification and the launch of high-energy weapons, the capacity level of a ship’s power system is increasing, and the rising rate of a short-circuit current can reach 20 A/μs. This puts forward high requirements for the rapidity of the protection device of the ship’s power system [1,2,3].

The action time of a traditional mechanical circuit breaker is generally at the millisecond level [4], which makes it difficult to meet the requirements. Solid state circuit breakers have the disadvantage of large rated current loss and needing additional heat dissipation devices [5,6,7].

An HDCCLCB combines the advantages of the large capacity of the mechanical switch and the rapidity of the solid-state switch, which becomes the development direction to solve the problem of fast DC breaking [8,9,10,11,12]. An HDCCLCB is generally composed of a high-speed mechanical switch, a metal oxide varistor (MOV), and a commutation circuit. Its basic principle is using the commutation circuit to provide the reverse current to realize the fast breaking of the circuit [13]. Therefore, the design of the commutation circuit directly determines the breaking speed of the circuit breaker [14,15,16].

To solve the problem of the fast breaking of the DC system, researchers have proposed many current commutation methods, optimized switch control strategies, and optimized commutation circuit parameters to achieve fast fault removal.

There are mainly three current commutation methods in HDCCLCBs. In reference [9,10], as long as the arc voltage of the mechanical switch is larger than the voltage drop of the commutation circuit, the current can be naturally commutated. However, limited by the arc voltage of the mechanical switch, this method can only be used in applications with a low current rising rate. In reference [13,17,18], to overcome current commutation difficulties in applications with a high current rising rate, the second method is proposed, in which the current is commutated by triggering the pre-charged LC oscillation circuit. In reference [19], to overcome the difficulty of the long dielectric recovery time of the mechanical switch when the rated current and the current rising rate of the ship’s DC system are high, the third method is proposed by combining the first two methods. Due to the high rising rate of the short-circuit current in the ship’s power system, the latter two methods are mainly used for breaking.

In addition to proposing different commutation methods, the control strategy of the commutation circuit also has a significant impact on the performance of the circuit breaker. In reference [20], when designing an HDCCLCB, the strategy of sending out the commutation circuit current in the opposite direction to the mechanical switch current after the mechanical contact has formed a certain opening distance is adopted. This design method of the commutation circuit is suitable for occasions where the requirements for current limiting are not high, and the current-limiting capacity is greatly affected by the moving speed of the mechanical switch. In reference [21,22], a new switch strategy for the optimization of a progressive switched hybrid DC circuit breaker is proposed and applied in the DC distribution system, which realized the low fault current peak and more uniform energy absorption of two MOV devices at the same time.

Researchers have also optimized the parameters of the commutation circuit. In reference [23], considering the influence of mechanical switch characteristics and the current-limiting inductance value on the breaking of the circuit breaker, the optimization design of the circuit breaker is carried out with the objective function of minimizing the number of semiconductors and the capacitor energy. In reference [24], the optimization design of the commutation circuit for the HDCCLCB is realized. According to Kirchhoff’s law, a genetic algorithm (GA) is used to calculate the commutation parameters under different optimization objectives to obtain the lowest cost and the shortest sheath development time. In reference [25], the absolute value of the product of the decline rate of the current before current zero (CZ) and the rising rate of the recovery voltage after CZ is used to measure the arc extinguishing capacity of the circuit breaker. Moreover, the product of the commutation capacitance value and the square of its rated voltage is used to evaluate the cost of the HDCCLCB. And the post-arc vacuum dielectric recovery characteristics are considered in sheath development based on the continuous transition model. The most reliable breaking operation, the lowest cost, and the shortest sheath development time are taken as the optimization objectives. Using a GA, the optimization of the commutation circuit parameters is proposed. The optimization results provide a reference for selecting the commutation inductance and commutation capacitor, according to the commutation circuit design for different practical conditions and different requirements. In reference [26], the breaker arc is modeled using the modified Mayr equations, and the absolute value of the product of the decline rate of the current before CZ and the rising rate of the recovery voltage after CZ is used to measure the arc extinguishing capacity of the circuit breaker. With the arc extinguishing capacity of the circuit breaker taken as the optimization objective, an iterative methodology is developed to determine the most suitable elements of the commutation circuit.

In reference [27,28,29], according to the requirement of fast DC breaking of the ship’s power system, a new topology of paralleling the reverse diode at two ends of the vacuum switch is proposed, and the control strategy of simultaneously generating a commutation current at the moment of vacuum switch opening is adopted. Through the reverse parallel diode, the vacuum switch can obtain the dielectric recovery time of zero voltage after the contact current flowing through zero so that the high-speed vacuum switch can break the high rising rate short-circuit current at a minimum opening distance.

For the new topology of the HDCCLCB in reference [27,28,29], the product of the decline rate of the current before CZ and the rising rate of the recovery voltage after CZ cannot be used to estimate the interrupting capacity of the circuit breaker. In this paper, aimed at the optimization design of commutation circuit parameters for the new topology, by simplifying the object, a preliminary parameter selection model considering the short-time withstand of the thyristor and the volume of the commutation circuit was built, and on this basis, the preliminary commutation circuit parameters were obtained. In order to further optimize the commutation circuit, a mathematical model for the optimization design of the commutation circuit is established. Taking the maximum critical breakdown voltage under unit capacitance energy as the objective function, the arcing time before commutating, the current-limiting inductance, and the capacitance are enumerated to obtain the optimized commutation circuit parameters. By comparing the optimized result with the result of the preliminary design, the objective function is improved by 20.7%, which verifies the effectiveness of the optimization model, laying a solid foundation for further research and the development of current-limiting circuit breakers for medium-voltage DC systems.

2. Operating Principle of the HDCCLCB

The circuit topology of the HDCCLCB is shown in Figure 1. The HDCCLCB is mainly composed of a high-speed vacuum interrupter (VI), a power electronic commutation circuit, a metal oxide varistor (MOV), and a reverse parallel diode (D). Its working process is as follows:

(1)

Under normal working conditions, the main circuit current flows through the vacuum interrupter (VI);

(2)

When the current needs to be cut off, the vacuum interrupter (VI) is opened, and the vacuum arc appears;

(3)

After a short arcing time, the thyristor (T) of the commutation circuit is triggered, and the pre-charged capacitor (C) discharges the vacuum interrupter (VI) through the inductor (L). The reverse parallel diode (D) cannot be turned on due to the clamping effect of the arc voltage, so the entire pulse current flows to the VI. The pulse current is opposite to the main circuit current, which makes the current in the VI gradually decrease until the current zero crossing is formed;

(4)

Because the reverse pulse current flows through the reverse parallel diode (D) after the arc is extinguished, the voltage drop on the contact of the VI is approximately zero. The transient recovery overvoltage will not occur until the pulse current is equal to the main circuit current again. During this phase, the VI enters the zero-voltage recovery time. In this process, the vacuum interrupter contact is still in motion, and the contact opening distance increases continuously, which is more conducive to the successful breaking of the VI.

3. Equivalent Circuit and Preliminary Parameter Design in the Commutation Process

3.1. Equivalent Circuit in the Commutation Process

Figure 2 shows the typical waveform of the current and voltage during the operation of the circuit breaker. According to the working principle of the circuit breaker, the equivalent circuit analysis is carried out for each phase of operation.

$0\to t_2$: At 0, the short-circuit fault occurs in the system. At $t_1$, when the current reaches the circuit breaker protection setting value $I_1$, the VI is triggered. After the inherent time $t_{\mathrm{g}}$ ($t_{\mathrm{g}}=t_2-t_1$) of the VI, the movable contact opens at t₂. At the $0\to t_2$ phase, the entire short-circuit current flows through the mechanical contact of the VI, and the VI is in the closing state. The circuit breaker can be equivalent to the resistance and inductance of the VI branch in series. The equivalent circuit diagram is shown in Figure 3. The thyristor conduction voltage drop is ignored in the equivalent circuit.

$t_2\to t_3$: At t₂, the VI contact opens, and there is an arc between the contacts, and the arc voltage is $U_{\mathrm{arc}}$. At t₃, the commutation circuit is turned on. Define the contact arcing time as $t_{\mathrm{r}}$ ($t_{\mathrm{r}}=t_3-t_2$). The equivalent circuit diagram of this phase is shown in Figure 4.

$t_3\to t_4$: At t₃, the commutation circuit is turned on to generate a pulse current. Due to the clamping effect of the arc voltage of the VI, the reverse parallel diode D is not turned on. With the increase in the pulse current, the current of the VI decreases gradually. At t₄, the current of the VI drops to zero, the current is completely commutated to the commutation circuit, and the commutation time is $t_{\mathrm{h}}$ ($t_{\mathrm{h}}=t_4-t_3$). The equivalent circuit at this phase is shown in Figure 5, where C is the commutation capacitor, U is the pre-charge voltage, R is the resistance, and L is the inductance of the commutation circuit.

$t_4\to t_5$: At t₄, the current of the VI decreases to zero, the vacuum arc is extinguished, and the reverse parallel diode D starts to be turned on. As the current of the commutation circuit continues to increase, the current of the commutation circuit commutates to diode D. Until the current of the commutation circuit drops to the current of the main circuit again, the reverse parallel diode D is cut off, and a recovery voltage starts to appear at two ends of the VI. The conduction time of the diode is the recovery time $t_{\mathrm{L}}$ ($t_{\mathrm{L}}=t_5-t_4$) of the VI. The equivalent circuit diagram of this phase is shown in Figure 6.

3.2. Preliminary Design of the Commutation Circuit Parameters

The design of the commutation circuit parameters mainly involves the $t_2\to t_5$ phase. The $t_2\to t_3$ phase determines the arcing time t_r before commutation, which together with the following $t_3\to t_4$ phase determines the arcing energy Q of the contact. The $t_4\to t_5$ phase determines the zero-voltage time t_L of the contact. According to the working principle of the circuit breaker, the parameter design of the commutation circuit shall meet the following conditions:

(1)

The peak value of the commutation circuit current I₆ shall be greater than the corresponding main circuit current I_max;

(2)

In order to meet the need of dielectric recovery of the VI, zero-voltage time $t_{\mathrm{L}}\ge 150\mathsf{\mu }\mathrm{s}$;

(3)

The currents of the commutation circuit shall all flow through the VI to reduce the energy stored in the capacitor;

(4)

Due to the limitation of the short-time withstand of the thyristor, the initial rising rate of the current in the commutation circuit $di/dt\le 220\mathrm{A}/\mathsf{\mu }\mathrm{s}$.

When ignoring the resistance R of the commutation circuit, the resistance of the VI, the inductance of the VI, the conduction voltage drop of the diode, and the arc voltage, the commutation circuit can be simplified as a second-order circuit in the zero state. The current-limiting resistance R is small which can be ignored. To further simplify the problem, assume that the cycle T of the second-order circuit is a constant k multiple of the zero-voltage time t_L, that is $k{t}_{\mathrm{L}}=T$. The following equations can be obtained:

$$U\sqrt{\frac{C}{L}}>I_{\max }$$

(1)

$$2\pi \sqrt{LC}>T$$

(2)

$$\frac{U}{L}<di/dt$$

(3)

Ignoring the resistance of the commutation circuit R and the resistance of the vacuum switch R_VI, the equivalent circuit diagram of the commutation process is shown in Figure 7, where i is the current of the commutation circuit, i_D is the current of the reverse parallel diode branch, and i_f is the reverse current flowing to the VI.

List circuit equations:

$$L_{\mathrm{D}}\frac{d{i}_{\mathrm{D}}}{dt}+u_{\mathrm{D}}+u_{\mathrm{arc}}-L_{\mathrm{VI}}\frac{d{i}_f}{dt}=0$$

(4)

It can be determined that the ratio of the current change rate of the current flowing to the VI and the reverse parallel diode circuit is:

$$\frac{d{i}_{\mathrm{D}}/dt}{d{i}_f/dt}=\frac{L_{\mathrm{VI}}-\frac{u_{\mathrm{D}}+u_{\mathrm{arc}}}{d{i}_f/dt}}{L_{\mathrm{D}}}$$

(5)

Whether the diode is turned on depends on the ratio of the current change rate. When the ratio of the current change rate is less than or equal to zero, the reverse parallel diode is not turned on, that is:

$$L_{\mathrm{VI}}-\frac{u_{\mathrm{D}}+u_{\mathrm{arc}}}{d{i}_f/dt}\le 0$$

(6)

Thus

$$\frac{d{i}_f}{dt}\le \frac{u_{\mathrm{D}}+u_{\mathrm{arc}}}{L_{\mathrm{VI}}}$$

(7)

When $i_{\mathrm{D}}$ is equal to zero, namely $i=i_f$, the constraint condition of the initial current rising rate di/dt of the commutation circuit can be obtained:

$$di/dt\le \frac{u_{\mathrm{D}}+u_{\mathrm{arc}}}{L_{\mathrm{VI}}}$$

(8)

The inductance value of the VI L_VI is 0.04 μH, the diode conduction voltage drop U_D is 1 V, and the arc voltage U_arc is 12 V; then:

$$di/dt\le 272\mathrm{A}/\mathsf{\mu }\mathrm{s}$$

(9)

Because the short-time withstand of the thyristor in the commutation circuit requires that the initial current rising rate is less than or equal to 220 A/μs, which is less than 272 A/μs, meeting the condition that the entire commutation current flows through the VI, the initial current rising rate of the commutation circuit should be less than 220 A/μs.

To sum up, each parameter needs to meet the following Equation (10): the capacitor charging voltage U is the system voltage 900 V and the current of the main circuit at the peak current of the commutation circuit is 15 kA, and the coefficient k is taken as 5 times. The parameter matching range is calculated through drawing, and the result is shown in Figure 8 and Figure 9:

$$\{\begin{array}{l}U/L<di/dt\\ U\sqrt{C/L}>I_{\max }\\ 2\pi \sqrt{LC}>T\end{array}\Rightarrow \{\begin{array}{l}L>U/(di/dt)\\ C>{(I_{\max }/U)}^{2}L\\ C>{[T/(2\pi )]}^2/L\end{array}$$

(10)

Point 1 is the intersection of curve 2 and 3; point 2 is the intersection of curve 1 and 3. The horizontal ordination of the two points are inductance L₁ and L₂, respectively.

$$\{\begin{array}{l}{L}_1=TU/\left(2\pi I_{\max }\right)\\ L_2=U/(di/dt)\end{array}$$

(11)

When L₁ is equal to L₂, that is $T=T_0$ = $2\pi I_{\max }/(di/dt)=429\mathsf{\mu }\mathrm{s}$.

(1)

When $T>T_0$, that is $t_{\mathrm{L}}>\frac{2\pi I_{\max }}{di/dtk}=85.8\mathsf{\mu }\mathrm{s}$, L₁ > L₂

The objective function is to minimize the volume of the commutation circuit. It is assumed that the larger the capacitance value is, the larger the capacitance volume is when the capacitance voltage is determined. At the same time, it is considered that the volume of the current-limiting inductance is incomparable with that of the capacitance. The optimization parameter matching is the parameter matching corresponding to point 1 in Figure 8, and the results are as follows:

$$\{\begin{array}{l}L=\frac{Uk{t}_{\mathrm{L}}}{2\pi I_{\max }}\\ C={(\frac{I_{\max }}{U})}^2\times L=\frac{I_{\max }k{t}_{\mathrm{L}}}{2\pi U}\\ C{U}^2=\frac{I_{\max }T}{2\pi U}\times U^2=\frac{I_{\max }k{t}_{\mathrm{L}}}{2\pi }U\end{array}$$

(12)

(2)

When $T\le T_0$, that is $t_{\mathrm{L}}\le \frac{2\pi I_{\max }}{di/dtk}=85.8\mathsf{\mu }\mathrm{s}$, L₁ ≤ L₂

The optimization parameter matching is the parameter matching corresponding to point 2 in Figure 9, and the results are as follows:

$$\{\begin{array}{l}L=\frac{U}{di/dt}\\ C={(\frac{I_{\max }}{U})}^2\times L=\frac{I_{\max }^2}{U\times di/dt}\\ C{U}^2=\frac{I_{\max }^2}{U\times di/dt}\times U^2=\frac{I_{\max }^{2}U}{di/dt}\end{array}$$

(13)

Under the assumption that the cycle of the commutation circuit is k times of the zero-voltage time and the optimization objective is to minimize the volume of the commutation circuit, the following conclusions are obtained:

(1)

When $t_{\mathrm{L}}>\frac{2\pi I_{\max }}{di/dtk}$, the matching of optimization parameters is: current-limiting inductance $L=\frac{Uk{t}_{\mathrm{L}}}{2\pi I_{\max }}$ and commutation capacitance $C=\frac{I_{\max }k{t}_{\mathrm{L}}}{2\pi U}$. Each parameter is independent of the initial current rising rate of the commutation circuit and proportional to the zero-voltage time.

(2)

When $t_{\mathrm{L}}\le \frac{2\pi I_{\max }}{di/dtk}$, the matching of optimization parameters is: current-limiting inductance $L=\frac{U}{di/dt}$ and commutation capacitance $C=\frac{I_{\max }^2}{Udi/dt}$. Each parameter is independent of the zero-voltage time and inversely proportional to the initial current rising rate of the commutation circuit.

Under the condition that the diode does not conduct during the commutation process, taking into account factors such as the short-time withstand of the thyristor, the volume of the commutation circuit, the capacitance of 2.0 mF, the pre-charge voltage of 900 V, and the current-limiting inductance of 7.5 μH are selected as the preliminary parameters.

4. Circuit Simulation Model of the HDCCLCB

The simulation model of the hybrid DC-current-limiting circuit breaker was built in circuit software, as shown in Figure 10.

The VI module is a vacuum switch module, and the vacuum arc voltage is assumed to be constant at 12 V, and the diode conduction voltage drop is 1 V.

Table 1. Breaking process schedule.

Time	Event
0 μs	Short circuit fault occurs
100 μs	Fault detection time $t_1=100\mathsf{\mu }\mathrm{s}$
220 μs	Inherent time of the vacuum interrupter $t_{\mathrm{g}}=120\mathsf{\mu }\mathrm{s}$
290 μs	Arcing time before commutating $t_{\mathrm{r}}=70\mathsf{\mu }\mathrm{s}$

shows the sequence of the breaking process.

At zero time, the thyristor T₀ is turned on, and the short-circuit fault occurs. The fault detection time is 100 μs, so the vacuum switch drive circuit is triggered at 100 μs. The inherent time of the vacuum switch is 120 μs, so at 220 μs, the switch starts to arc, forming a vacuum arc with an arc voltage of 12 V. After the arcing time of 70 μs, the thyristor T is triggered at 290 μs, and the commutation circuit starts to work.

5. Prototype Experiment and Circuit Model Verification

Figure 11 shows the structure diagram of the experimental platform. In the experiment, the capacitor bank is used as the system power supply to simulate the short-circuit fault. A Rogowski coil and a high-voltage probe are used to obtain the current and voltage information, respectively. As shown in Figure 12, the rising rate of the short-circuit current is 20 A/μs, and the waveforms of the main circuit current i0, contact voltage U_VI, contact current i_VI, and commutation circuit current i are recorded.

It can be seen from Figure 12 that the simulation and experimental current waveforms are almost identical. The reason why the voltage waveform (experimental value products 100) has a voltage value during the phase of 0~200 μs, which is only affected by the current rising rate of the main circuit, is that the voltage probe is not fully attached to the two ends of the vacuum contact during the measurement, including part of the main circuit inductance. The error between the contact voltage waveform and the experimental waveform during the commutation process is large, because the contact voltage is affected by the change in the contact current and contact opening distance during the commutation process, which is considered as a constant value in the simulation. Since the minimum voltage of the arc voltage is taken as the voltage calculation value when calculating the shunt ratio between the contact circuit and diode circuit, and the initial current rising rate has been limited, so there will be no diode shunt, this error does not affect the analysis of the commutation process, thus verifying the correctness of the circuit simulation model.

6. Optimization Design of the Commutation Circuit Parameters

The parameter design of the commutation circuit is the core task of the circuit breaker design. Therefore, the optimization mathematical model of the commutation circuit was established.

6.1. Optimization Analysis Model of the Commutation Circuit Parameters

The vacuum switch resistance, vacuum switch inductance, and arc voltage are ignored. In the $t_2\to t_4$ phase, the circuit can be equivalent to two zero-state second-order circuits. The equivalent circuit is shown in Figure 13.

List the circuit equation:

$$i_{\mathrm{VI}}=i_0-i$$

(14)

The main circuit:

$$\{\begin{array}{l}{L}_0{C}_0\frac{d^2{i}_0}{dt}+R_0{C}_0\frac{d{i}_0}{dt}+i_0=0\\ i_0(0_{+})=0\\ u_0(0_{+})=U_0\end{array}t\ge 0$$

(15)

The commutation circuit:

$$\{\begin{array}{l}i=0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}0\le t<t_3\\ LC\frac{d^{2}i}{d(t-t_3)}+RC\frac{di}{d(t-t_3)}+i=0t\ge t_3\\ i(0_{+})=0\\ u(0_{+})=U\end{array}$$

(16)

Simultaneously solve Equations (15) and (16) to obtain the solution between [$t_3$ $(t_3+T/2)$]. The arcing time before commutation t_r, commutation time t_h, and zero-voltage time t_L are respectively:

$$t_{\mathrm{r}}=t_3-t_2$$

(17)

$$t_{\mathrm{h}}=t_4-t_3$$

(18)

$$t_{\mathrm{L}}=t_5-t_4$$

(19)

Arcing energy Q:

$$Q={\displaystyle \int u_{arc}.i_{VB}}dt{t}_2\le t\le t_4$$

(20)

In this paper, the change in the voltage is not considered as temporary, and it is constantly 12 V. Reference [20] shows the relationship between the critical breakdown voltage and the zero-voltage time:

$$V_{\mathrm{b}}=U_{\mathrm{S}}(1-\exp [-t_{\mathrm{L}}k\mathrm{D}/Q])$$

(21)

where U_s is the cold breakdown voltage of the vacuum contact and D is the contact opening distance. In case of a short gap (less than 5 mm), the vacuum critical breakdown voltage is proportional to the contact opening distance, and the following equation can be obtained:

$$V_{\mathrm{b}}=Ev(t_{\mathrm{r}}+t_{\mathrm{h}}+t_{\mathrm{L}})(1-\exp [-kv{t}_{\mathrm{L}}(t_{\mathrm{r}}+t_{\mathrm{h}}+t_{\mathrm{L}})/Q])$$

(22)

where v is the contact movement speed. Define objective function y:

$$y=\frac{V_b}{C{U}^2}$$

(23)

The arcing time before commutating, the current-limiting inductance value, the capacitance value, and the initial voltage of the capacitance are enumerated to obtain the optimization parameter. The range of parameter selection is within three times of the results of the preliminary design. The optimized value can be obtained by eliminating the situation where there are no two different solutions, the short-time withstand of the thyristor is not satisfied, the initial rising rate of the commutation current is not satisfied, and the zero-voltage time is not satisfied.

6.2. Influence of the Parameters on the Objective Function

In this section, the influence of each parameter on the critical breakdown voltage and objective function is simulated and analyzed.

Figure 14 shows the influence of the arcing time before commutating on the contact movement time and arcing energy. With the increase in the arcing time, the contact movement time and arcing energy increase approximately linearly at the same time. The critical breakdown voltage will increase with the increase in the contact movement time, that is, the opening distance, but the critical breakdown voltage will decrease with the increase in the arcing energy. The two contradict each other.

Figure 15 shows the influence of the arcing time on the critical breakdown voltage and objective function. It can be seen from Figure 15 that the critical breakdown voltage and objective function have a maximum value with the increase in the arcing time, and the maximum value corresponds to the arcing time of 80 μs. It can be seen that when the arcing time is less than 80 μs, the critical breakdown voltage is mainly dominated by the contact opening distance, while when it is greater than 80 μs, the critical breakdown voltage is mainly dominated by the arcing energy.

Figure 16 shows the influence of the current-limiting inductance value L on the contact movement time and arcing energy. With the increase in the current-limiting inductance, the contact movement time firstly increases and then decreases, and there is a maximum value; the horizontal ordinate of the maximum value is 8.5 μH, while the arcing energy increases. The increase in the current-limiting inductance will increase the commutation time and decrease the zero-voltage time, so the movement time will increase firstly and then decrease. At the same time, as the commutation time increases, the arcing time before commutating remains unchanged, and the total arcing time increases, so the arcing energy increases.

Figure 17 shows the influence of the current-limiting inductance value L on the critical breakdown voltage and objective function. With the increase in the current-limiting inductance, the critical breakdown voltage and objective function firstly increase and then decrease, and there is a maximum value, and the horizontal ordinate of the maximum value is 8μH. It can be known by comparing the horizontal ordinate of the contact movement time and the maximum critical breakdown voltage. In the process of changing the current-limiting inductance, the critical breakdown voltage is mainly dominated by the contact movement time.

The influence of the capacitance C on the contact movement time and arcing energy is shown in Figure 18. With the increase in the capacitance, the contact movement time increases and the arcing energy decreases. The increase in the capacitance will reduce the commutation time and increase the zero-voltage time, but the zero-voltage time will increase more, so the contact movement time will increase. When the commutation time decreases, the arcing time before commutation remains unchanged, which will reduce the total arcing time, so the arcing energy decreases.

As shown in Figure 19, the influence of capacitance C on the critical breakdown voltage and the objective function is shown. The critical breakdown voltage increases with the increase in the capacitance. This is because the increase in the contact movement time and the decrease in the arcing energy are beneficial to the critical breakdown voltage, so the critical breakdown voltage increases. With the increase in capacitance, the objective function firstly increases and then decreases. When the capacitance is 1.4 mF, the objective function is the largest.

The influence of capacitance voltage U on the contact movement time and arcing energy is shown in Figure 20. With the increase in the capacitor voltage, the contact movement time increases and the arcing energy decreases. The increase in the capacitor voltage will reduce the commutation time and increase the zero-voltage time, but the zero-voltage time will increase more, so the contact movement time will increase. When the commutation time decreases, the arcing time before commutation remains unchanged, which will reduce the total arcing time, so the arcing energy decreases.

Figure 21 shows the influence of capacitor voltage U on the critical breakdown voltage and objective function. The critical breakdown voltage increases with the increase in the capacitor voltage. This is because the increase in the contact movement time and the decrease in the arcing energy are beneficial to the critical breakdown voltage, so the critical breakdown voltage increases. The objective function decreases with the increase in the capacitor voltage. This is because the capacitor energy is proportional to the square of the voltage. Increasing the capacitor voltage will lead to a sharp increase in energy consumption, so the objective function decreases.

6.3. Optimization Results and Simulation Comparison

In order to compare the results between the preliminary design and the optimization design, the system voltage U and the arcing time before commutating are consistent with the preliminary design. Taking the maximum critical breakdown voltage under unit capacitance energy as the objective function, current-limiting inductance and capacitance are enumerated to obtain the optimized commutation circuit parameters. The capacitance value is taken every 0.1 mF within the range of 0.8~3.2 mF, and the current-limiting inductance value is taken every 0.03 μH within the range of 5.5~9.5 μH.

Table 2. Parameter matching of the preliminary design and the optimization design.

	tr/μs	L/μH	C/mF	U/V
1	24	7.5	2	900
2	24	6.16	1.2	900

shows the parameter comparison between the preliminary design and the optimization design. Numbers 1 and 2 represent the preliminary design and optimization design, respectively. Figure 22 shows the waveform of the simulation current.

It can be seen from Figure 22 that under optimization parameter matching, the arcing time is 24 μs, the commutation time is 41 μs, and the zero-voltage time is 161 μs, which meets the design requirements.

Table 3. Comparison between the preliminary design and the optimization design.

Parameter	Preliminary Design	Optimization Design	Percentage Difference
di/dt/(A/μs)	120	146	+21.7%
Q/J	3.52	3.14	−10.8%
CU2/J	1620	972	−40%
Vb/V	18,955	13,733	−27.5%
y	11.70	14.12	+20.7%

shows the comparison between the preliminary design and optimization design. It can be seen from Table 3 that although the critical breakdown voltage is reduced by 27.5% by reducing the capacitor energy after optimization, the objective function is therefore increased by 20.7%. This proves the correctness of the optimization method.

7. Discussion

The sixth section considers the impact of the current-limiting inductance and capacitance values on the objective function, without considering the impact of the arcing time before commutation and capacitance voltage. From Figure 14 and Figure 15, it can be seen that under the same parameters of the commutation circuit, increasing the arcing time before commutation will not only increase the opening distance of the contact but also increase the arcing energy, and the critical breakdown voltage and objective function have a maximum value with the increase in the arcing time. Is there an extreme value under different current-limiting inductance and capacitance values? From Figure 20 and Figure 21, it can be seen that when only the capacitor voltage is changed, the capacitor voltage is inversely proportional to the objective function. If we consider the influence of other parameters, is it still true that the smaller the capacitor voltage, the larger the objective function? Therefore, the objective function is optimized by comprehensively considering the capacitance voltage, the current-limiting inductance value, the capacitance value, and the arcing time before commutation. The capacitance voltage is taken every 225 V within the range of 225~1800 mF, the capacitance value is taken every 0.5 mF within the range of 0.5~4 mF, the current-limiting inductance value is taken every 1.875 μH within the range of 1.875~15 μH, and the arcing time is taken every 20 μs within the range of 20~160 μs.

Table 4. The optimal parameter combinations under different conditions.

	No Limitations on di/dt and tL	Only Considering the Limitation on tL	Only Considering the Limitation on di/dt	Considering the Limitations on di/dt and tL
tL/μs	19.45	152 > 150	17.76	156 > 150
di/dt/(A/μs)	360	240	180 < 220	120 < 220
L/μH	1.875	1.875	3.75	3.75
C/mF	0.5	4	1	4
U/V	675	450	675	450
tr/μs	160	140	140	60
Q/J	19.37	17.43	18.69	9.54
CU2/J	228	810	456	810
Vb/V	12,900	20,100	14,100	17,900
y	56.58	24.81	30.92	22.10

shows the optimal parameter combinations under different conditions.

According to Table 4, the optimal parameter combination not considering the limitations on di/dt and t_L is L = 1.875 μH, C = 0.5 mF, U = 675 V, and t_r = 160 μs, which shows that within a certain parameter range, when the arcing time is taken as the maximum value, the current-limiting inductance value is taken as the minimum value, the capacitance value is taken as the minimum value, and the capacitance voltage is taken as the value that can ensure that the pulse current is greater than the main circuit current, which can maximize the objective function value. When only considering the limitation on t_L, increasing the capacitance value is better than increasing the capacitance voltage value, as the capacitance energy is proportional to the square of the capacitance voltage. When only considering the limitation on di/dt, increasing the current-limiting inductance value is better than decreasing the capacitance voltage value, as the peak current is proportional to the voltage and inversely proportional to the half power of inductance. When considering the limitations on di/dt and t_L, the optimal parameter combination is L = 3.75 μH, C = 4 mF, U = 450 V, and t_r = 60 μs.

8. Conclusions

Aimed at the optimization design of the commutation circuit parameters of a hybrid DC-current-limiting circuit breaker, this paper presents an optimization design method and verifies the optimization results through experiments. It obtains the following conclusions:

(1)

Under the condition of only changing the arcing time, with the increase in the arcing time, the critical breakdown voltage and the objective function of the contact firstly increase and then decrease, and there is a maximum value;

(2)

Under the condition of only changing the current-limiting inductance, with the increase in the current-limiting inductance, the critical breakdown voltage and the objective function of the contact firstly increase and then decrease, and there is a maximum value;

(3)

Under the condition of only changing the capacitance value, the critical breakdown voltage of the contact always increases with the increase in the capacitance value. The objective function firstly increases and then decreases, and there is a maximum value;

(4)

Under the condition of only changing the capacitor voltage, the critical breakdown voltage of the contact increases with the increase in the capacitor voltage. The objective function always decreases;

(5)

By comprehensively considering the impact of various factors on the objective function rather than solely considering the impact of a single factor, further optimization of the objective function can be achieved;

(6)

The optimization design model established in this paper can optimize the parameters of the commutation circuit, laying a solid foundation for further analysis and the design of medium-voltage DC-current-limiting circuit breakers for ships.